There are certainly some similarities between LOCALLY GROWN and FRANKLIN D. ROOSEVELT'S BACK AGAIN. The metrical structure is nearly the same, and maybe even the chords are the same, but enough notes are different that I don't think you can say it's the same tune.

If Chapin did borrow a tune and modify it or adapt it, whether consciously or unconsciously, he's only doing what hundreds of folk singers have done before him.

By the way, since you're editing this for publication, I suggest that you either put every rhyming word at the end of a line like this:

If I were an apple
I'd be very unhapple
Traveling four thousand miles or more
From far off Tasmania,
In a shipping contain-ia
To a shelf in a New Jersey store.
Why should I be tortured
When some New Jersey orchard
Would be totally thrilled to the core
To pick me and crate me
And load me and freight me
Not four thousand miles, but four?
An apple should be
Not far from the tree
Where it ripens in the fall. [etc,]

Or, if you prefer internal rhyme rather than lots of short lines, use it consistently, like this:

If I were an apple I'd be very unhapple
Traveling four thousand miles or more
From far off Tasmania, in a shipping contain-ia
To a shelf in a New Jersey store.
Why should I be tortured when some New Jersey orchard
Would be totally thrilled to the core
To pick me and crate me and load me and freight me
Not four thousand miles, but four?
An apple should be not far from the tree
Where it ripens in the fall. [etc,]

Regardless, every verse should have the same number of lines, and the line breaks should come at places that are consistent from one verse to the next.

(The boldface is just to show you what I mean; I don't think you should use boldface in the publication.)